Question

find the perimeter of the triangle with these vertices. (-3, 5), (2, 1), (-3, 1) give an exact answer (not a decimal approximation). simplify your answer as much as possible. perimeter:

Explanation:

Step1: Calculate side - 1 length

Use distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $(-3,5)$ and $(-3,1)$, $x_1=-3,y_1 = 5,x_2=-3,y_2 = 1$. Then $d_1=\sqrt{(-3+3)^2+(1 - 5)^2}=\sqrt{0+( - 4)^2}=4$.

Step2: Calculate side - 2 length

For points $(-3,1)$ and $(2,1)$, $x_1=-3,y_1 = 1,x_2=2,y_2 = 1$. Then $d_2=\sqrt{(2 + 3)^2+(1 - 1)^2}=\sqrt{5^2+0}=5$.

Step3: Calculate side - 3 length

For points $(2,1)$ and $(-3,5)$, $d_3=\sqrt{( - 3 - 2)^2+(5 - 1)^2}=\sqrt{( - 5)^2+4^2}=\sqrt{25 + 16}=\sqrt{41}\approx6.4$. But we can also use the Pythagorean theorem for the right - triangle formed. The horizontal distance is $5$ and the vertical distance is $4$, so $d_3=\sqrt{5^2+4^2}= \sqrt{25+16}= \sqrt{41}\approx6.4$. The perimeter $P=d_1 + d_2+d_3=4 + 5+3=12$.

Answer:

$12$